In order to distinguish the different lenses in this eye-piece, we may call the lens A, which is next to the first image, the _object-lens_, the next to it B, the _amplifying-lens_, the third, or C, the _field-lens_, and the one next the eye, D, the _eye-lens_. The first image formed a little before A, may be denominated the _radiant_, or the object from which the rays proceed. Now, it is well known as a principle in optics, that if the radiant be brought nearer to the lens than its principal focus, the emerging rays will _diverge_, and, on the contrary, if the radiant be put farther from the lens than its principal focal distance, the emerging rays will _converge_ to a point at a distance beyond the lens, which will depend on the distance of the radiant from the first face of the lens. In this place an image of the radiant will be formed by the concurrence of the converging rays, but in a contrary position; and the length of the image will exceed the length of the radiant in the same proportion, as the distance of the image from the radiant exceeds that of the radiant from the lens. This secondary image of the radiant at _i_, is not well-defined, when only one lens, as A, is used, owing to the great spherical aberrations, and therefore the amplifying lens is placed at the distance of the shorter conjugate focus, with an intervening diaphragm of a small diameter at the place of the principal focus; the uses of which lens and diaphragm are, first to cut off the coloured rays that are occasioned by the dispersive property of the object lens,--and secondly, to bring the rays to a shorter conjugate focus for the place of the image, than would have taken place with a single lens having only one refraction. As the secondary image is in this way much better defined and free from colouration, the addition of this second lens is a great improvement to vision. For this reason I am clearly of opinion, that the object glass of a compound microscope, instead of consisting of a small single lens, should be formed of two lenses on the principle now stated, which would unquestionably add to the distinctness of vision. With respect to _the proportions of the focal lengths of the lenses_ in this four glass eye-piece, Mr. Coddington states, that if the focal lengths, reckoning from A to D, fig. 79, be as the numbers 3, 4, 4 and 3, and the distances between them on the same scale, 4, 6, and 5, 2, the radii, reckoning from the outer surface of A, should be thus:-- A {First surface 27 }nearly plano-convex. {Second surface 1 } B {First surface 9 }a miniscus. {Second surface 4 } C {First surface 1 }nearly plano-convex. {Second surface 21 } D {First surface 1 }double convex. {Second surface 24 }